Performance Analysis and Nash Equilibria in a Taxi-Passenger System with Two Types of Passenger

Abstract

This paper analyzes a taxi-passenger queueing system where there are two types of passengers differentiated by their mean matching times with taxis. System states can be observable or unobservable to agents. Performance measures of the system are derived using a Markov chain with three-dimensional states. Furthermore, we consider the case where multiple types of agents are strategic and use a multi-population game theoretical framework to analyze the system in equilibrium.

Appendix A Definition of Block Matrices in Section “Performance Measures”

$$\begin{aligned}&{\mathcal {A}}^{(K)},{\mathcal {B}}^{(K)},{\mathcal {C}}^{(K)} \in {\mathbb {M}}\nonumber \\&\left( \frac{(S+1)(S+2)}{2}+(K-S)(S+1), \frac{(S+1)(S+2)}{2}+(K-S)(S+1) \right) \end{aligned}$$

such that

$$\begin{aligned}&{\mathcal {C}}^{(K)}=diag(\lambda ,\lambda ,...,\lambda );\\&{\mathcal {A}}^{(K)}\left( \frac{(S+1)(S+2)}{2}+i(S+1)+j,\frac{S(S+1)}{2}+i(S+1)+j \right) \\&=\alpha (j-1)\mu _1+(1-\alpha )(S-(j-1))\mu _2,\\&{\mathcal {A}}^{(K)}\left( \frac{(S+1)(S+2)}{2}+i(S+1)+j, \frac{S(S+1)}{2}+i(S+1)+(j+1) \right) \\&=\alpha (S-(j-1))\mu _2,\\&{\mathcal {A}}^{(K)}\left( \frac{(S+1)(S+2)}{2}+i(S+1)+(j+1),\frac{S(S+1)}{2}+i(S+1)+j \right) \\&=(1-\alpha )j\mu _1,\\&{\mathcal {B}}^{(K)}\left( \frac{S(S+1)}{2}+i(S+1)+j,\frac{(S+1)(S+2)}{2}+i(S+1)+(j+1) \right) =\lambda _t, \end{aligned}$$

for $i=0,1,...,K-S$ and $j=1,2,...,S+1;$

$$\begin{aligned}&{\mathcal {A}}^{(K)}\left( \frac{(i+1)(i+2)}{2}+j,\frac{i(i+1)}{2}+j \right) =(i-(j-1))\mu _2, \\&{\mathcal {A}}^{(K)}\left( \frac{(i+1)(i+2)}{2}+(j+1),\frac{i(i+1)}{2}+j \right) =j\mu _1, \\&{\mathcal {B}}^{(K)}\left( \frac{i(i+1)}{2}+j,\frac{(i+1)(i+2)}{2}+j \right) =(1-\alpha )\lambda _t, \\&{\mathcal {B}}^{(K)}\left( \frac{i(i+1)}{2}+j,\frac{(i+1)(i+2)}{2}+(j+1) \right) =\alpha \lambda _t, \end{aligned}$$

for $i=0,1,...,S-1$ and $j=1,2,...,i+1.$

For $n<K$,

$$\begin{aligned}&{\mathcal {C}}^{(n)} \in {\mathbb {M}}\\&\left( \frac{(n+1)(n+2)}{2}+(K-n)(n+1),\frac{(n+1)(n+2)}{2}+(K-n)(n+2) \right) , \end{aligned}$$

for $n \ge 1$, and

$$\begin{aligned}&{\mathcal {A}}^{(n)} \in {\mathbb {M}}\left( \frac{(n+1)(n+2)}{2}+(K-n)(n+1),\frac{n(n+1)}{2}+(K-(n-1))n \right) ,\\&{\mathcal {B}}^{(n)} \in {\mathbb {M}}\left( \frac{(n+1)(n+2)}{2}+(K-n)(n+1),\frac{(n+1)(n+2)}{2}+(K-n)(n+1) \right) , \end{aligned}$$

such that

$$\begin{aligned}&{\mathcal {C}}^{(n)}(i,i)=\lambda \end{aligned}$$

for $i=1,2,...,\frac{(n+1)(n+2)}{2};$

$$\begin{aligned}&{\mathcal {C}}^{(n)}\left( \frac{(n+1)(n+2)}{2}+i(n+1)+j,\frac{(n+1)(n+2)}{2}+i(n+2)+j \right) \\&=(1-\alpha )\lambda ,\\&{\mathcal {C}}^{(n)}\left( \frac{(n+1)(n+2)}{2}+i(n+1)+j,\frac{(n+1)(n+2)}{2}+i(n+2)+(j+1) \right) \\&=\alpha \lambda ,\\&{\mathcal {A}}^{(n)}\left( \frac{(n+1)(n+2)}{2}+i(n+1)+j,\frac{n(n+1)}{2}+i(n+1)+j \right) \\&=(n-(j-1))\mu _2,\\&{\mathcal {A}}^{(n)}\left( \frac{(n+1)(n+2)}{2}+i(n+1)+(j+1),\frac{n(n+1)}{2}+i(n+1)+j \right) \\&=j\mu _1,\\&{\mathcal {B}}^{(n)}\left( \frac{n(n+1)}{2}+i(n+1)+j,\frac{(n+1)(n+2)}{2}+i(n+1)+(j+1) \right) \\&=\lambda _t, \end{aligned}$$

for $i=0,1,...,K-n$ and $j=1,2,...,n+1;$

$$\begin{aligned}&{\mathcal {A}}^{(n)}\left( \frac{(i+1)(i+2)}{2}+j,\frac{i(i+1)}{2}+j \right) \\&=(i-(j-1))\mu _2,\\&{\mathcal {A}}^{(n)}\left( \frac{(i+1)(i+2)}{2}+(j+1),\frac{i(i+1)}{2}+j \right) \\&=j\mu _1,\\&{\mathcal {B}}^{(n)}\left( \frac{i(i+1)}{2}+j,\frac{(i+1)(i+2)}{2}+j \right) \\&=(1-\alpha )\lambda _t,\\&{\mathcal {B}}^{(n)}\left( \frac{i(i+1)}{2}+j,\frac{(i+1)(i+2)}{2}+(j+1) \right) \\&=\alpha \lambda _t, \end{aligned}$$

for $i=0,1,...,n-1$ and $j=1,2,...,i+1;$ and $n \ge 1$. Finally,

$$\begin{aligned}&\mathcal{B}^{(0)}(i,i) = -\sum _{j}\mathcal{A}^{(0)}(i,j) -\sum _{j \ne i}\mathcal{B}^{(0)}(i,j), \end{aligned}$$

and

$$\begin{aligned}&\mathcal{B}^{(n)}(i,i) = -\sum _{j}\left( \mathcal{A}^{(n)}(i,j)+\mathcal{C}^{(n)}(i,j)\right) -\sum _{j \ne i}\mathcal{B}^{(n)}(i,j) \end{aligned}$$

for $n=1,...,K.$